Method for estimating a position of a wellbore

ABSTRACT

A method is disclosed which utilizes multiple overlapping surveys to estimate a position in a wellbore and related position uncertainty. Multiple surveys are often taken over the same portion of a wellbore either concurrently or sequentially and/or using various instruments. Each survey generates an estimated survey position and related uncertainty for a given location in the wellbore. By combining the estimated survey positions and uncertainties for these overlapping surveys, a resultant position and related ellipsoid of uncertainty is estimated. This resultant position estimates a position in the wellbore by incorporating the estimated survey positions and uncertainties of multiple overlapping surveys.

BACKGROUND OF INVENTION

1. Field of the Invention

The invention relates generally to wellbore surveys. More particularly,the invention relates to the estimation of wellbore positions based onanalytical techniques.

2. Background Art

Fluids, such as oil, gas and water, are commonly recovered fromsubterranean formations below the earth's surface. Drilling rigs at thesurface are often used to bore long, slender wellbores into the earth'scrust to the location of the subsurface fluid deposits to establishfluid communication with the surface through the drilled wellbore. Thelocation of subsurface fluid deposits may not be located directly(vertically downward) below the drilling rig surface location. Awellbore that defines a path, which deviates from vertical to somelaterally displaced location, is called a directional wellbore. Downholedrilling equipment may be used to directionally steer the wellbore toknown or suspected fluid deposits using directional drilling techniquesto laterally displace the borehole and create a directional wellbore.

The path of a wellbore, or its “trajectory,” is made up of a series ofpositions at various points along the wellbore obtained by using knowncalculation methods. “Position,” as the term is used herein, refers toan orthogonal Cartesian (x, y, z) spatial position, referenced to somevertical and/or horizontal datum (usually the well-head position andelevation reference). The position may also be obtained using inertialmeasurement techniques, or by using inclination and azimuth with knowncalculation methods. “Azimuth” may be considered, for present purposes,to be the directional angular heading, relative to a referencedirection, such as North, at the position of measurement. “Inclination”may be considered, also for present purposes, to be the angulardeviation from vertical of the borehole at the position of measurement.

Directional wellbores are drilled through earth formations along aselected trajectory. Many factors may combine to unpredictably influencethe intended trajectory of a wellbore. It is desirable to accuratelyestimate the wellbore trajectory in order to guide the wellbore to itsgeological and/or positional objective. This makes it desirable tomeasure the inclination, azimuth and depth of the wellbore duringwellbore operations to estimate whether the selected trajectory is beingmaintained.

The drilled trajectory of a wellbore is estimated by the use of awellbore or directional survey. A wellbore survey is made up of acollection or “set” of survey-stations. A survey station is generated bytaking measurements used for estimation of the position and/or wellboreorientation at a single position in the wellbore. The act of performingthese measurements and generating the survey stations is termed“surveying the wellbore.”

Surveying of wellbores is commonly performed using downhole surveyinstruments. These instruments typically contain sets of orthogonalaccelerometers, magnetometers and/or gyroscopes. These surveyinstruments are used to measure the direction and magnitude of the localgravitational, magnetic field and/or earth spin rate vectorsrespectively, herein referred to as “earth's vectors”. Thesemeasurements correspond to the instrument position and orientation inthe wellbore, with respect to earth vectors. Wellbore position,inclination and/or azimuth may be estimated from the instrument'smeasurements.

One or more survey stations may be generated using “discrete” or“continuous”measurement modes. Generally, discrete or “static” wellboresurveys are performed by creating survey stations along the wellborewhen drilling is stopped or interrupted to add additional joints orstands of drillpipe to the drillstring at the surface. Continuouswellbore surveys relate to thousands of measurements of the earth'svectors and/or angular velocity of a downhole tool obtained for eachwellbore segment using the survey instruments. Successive measurementsof these vectors during drilling operations may be separated by onlyfractions of a second or thousandths of a meter and, in light of therelatively slow rate of change of the vectors in drilling a wellbore,these measurements are considered continuous for all practical analyses.

Known survey techniques as used herein encompass the utilization of avariety of means to estimate wellbore position, such as using sensors,magnetometers, accelerometers, gyroscopes, measurements of drill pipelength or wireline depth, Measurement While Drilling (“MWD”) tools,Logging While Drilling (“LWD”) tools, wireline tools, seismic data, andthe like.

Surveying of a wellbore is often performed by inserting one or moresurvey instrument into a bottom-hole-assembly (“BHA”), and moving theBHA into or out of the wellbore. At selected intervals, usually aboutevery 30 to 90 feet (10 to 30 meters), BHA, having the instrumenttherein, is stopped so that measurement can be made for the generationof a survey station. An additional measurement not performed by thesurvey instruments is the estimation of the along hole depth (measureddepth “MD”) or wellbore distance between discrete survey stations. TheMD corresponds to the length of joints or stands of drillpipe added atthe surface down to the BHA survey station measurement position. Themeasurements of inclination and azimuth at each survey station alongwith the MD are then entered into any one of a number of well-knownposition calculation models to estimate the position of the surveystation to further define the wellbore trajectory up to that surveystation.

Existing wellbore survey computation techniques use various models,including the Tangential method, Balanced Tangential method, AverageAngle method, Mercury method, Differential Equation method, cylindricalRadius of Curvature method and the Minimum Radius of Curvature method,to model the trajectory of the wellbore segments between surveystations.

Directional surveys may also be performed using wireline tools. Wirelinetools are provided with one or more survey probes suspended by a cableand raised and lowered into and out of a wellbore. In such a system, thesurvey stations are generated in any of the previously mentionssurveying modes to create the survey. Often wireline tools are used tosurvey wellbores after a drilling tool has drilled a wellbore and an MWDand/or LWD survey has been previously performed.

Uncertainty in the survey results from measurement uncertainty, as wellas environmental factors. Measurement uncertainty may exist in any ofthe known survey techniques. For example, magnetic measuring techniquessuffer from the inherent uncertainty in global magnetic models used toestimate declination at a specific site. Similarly, gravitationalmeasuring techniques suffer from movement of the downhole tool anduncertainties in the accelerometers. Gyroscopic measuring techniques,for example, suffer from drift uncertainty. Depth measurements are alsoprone to uncertainties including mechanical stretch from gravitationalforces and thermal expansion, for example.

Various considerations have brought about an ever-increasing need formore precise wellbore surveying techniques. More accurate surveyinformation is necessary to ensure the avoidance of well collisions andthe successful penetration of geological targets.

Surveying techniques have been utilized to estimate the wellboreposition. For example, techniques have also been developed to estimatethe position of wellbore instruments downhole. U.S. Pat. No. 6,026,914to Adams et al. relates to a wellbore profiling system utilizingmultiple pressure sensors to establish the elevation along the wellborepath. U.S. Pat. No. 4,454,756 to Sharp et al. relates to an inertialwellbore survey system, which utilizes multiple accelerometers, andgyros to serially send signals uphole. U.S. Pat. No. 6,302,204 B1 toReimers et al. relates to a method of conducting subsurface seismicsurveys from one or more wellbores from a plurality of downhole sensors.U.S. Pat. No. 5,646,611 to Dailey et al. relates to the use of twoinclinometers in a drilling tool to estimate the inclination angle ofthe wellbore at the bit.

Other techniques have been developed to correct data based onmeasurement error. U.S. Pat. No. 6,179,067 B1 to Brooks relates to amethod for correcting measurement errors during survey operations bycorrecting observed data to a model. U.S. Pat. No. 5,452,518 to DiPersiorelates to a method of estimating wellbore azimuth by utilizing aplurality of estimates of the axial component of the measured magneticfield by emphasizing the better estimates and de-emphasizing poorerestimates to compensate for magnetic field biasing error.

There remains a need for techniques capable of utilizing overlappingsurvey data to better estimate the wellbore position and its relateduncertainty of that position. Mathematical models have been used toestimate the wellbore position and position uncertainty in a wellbore.For example, SPE 56702 entitled “Accuracy Prediction for DirectionalMWD,” by Hugh S. Williamson (©1999), SPE 9223 entitled “BoreholePosition Uncertainty, Analysis of Measuring Methods and Derivation ofSystematic Error Model,” by Chris J. M. Wolff and John P. De Wardt(©1981), and “Accuracy Prediction for Directional Measurement WhileDrilling,” by H. S. Williamson, SPE Drill and Completion, Vol. 15, No. 4Dec. 2000, the entire contents of which are hereby incorporated byreference, describe mathematical techniques used in wellbore positionanalysis. However, a specific position in a wellbore is often surveyedmany times and by many different types of survey instruments at variousstages of wellbore operations. Historically, these existing methods relyupon a sequence of non-overlapping surveys along the wellbore toestimate the position of a point in the wellbore, and fail toincorporate overlapping survey data.

It is desirable that overlapping surveys be taken into considerationwhen estimating positions in a wellbore. It is also desirable that amethod of estimating positions in the wellbore, use overlapping surveysgenerated by downhole tools. The present invention provides a technique,which utilizes multiple overlapping surveys and combines the overlappingsurveyed positions and related positional uncertainties of a givenwellpath in order to produce a resultant wellbore position, or ‘MostProbable Position’ (MPP), as well as an associated resultant positionaluncertainty.

SUMMARY OF INVENTION

An aspect of the invention relates to a method for estimating a positionin a wellbore. The method involves acquiring a plurality of surveys ofthe wellbore and combining overlapping portions of the surveys wherebythe wellbore position is determined. Each measured survey defines asurvey position in the wellbore and an uncertainty of the surveyposition.

Another aspect of the invention relates to a method for estimating aposition in a wellbore. The method involves drilling a wellbore into asubterranean formation, acquiring a plurality of surveys of the wellboreand combining overlapping portions of the surveys whereby the wellboreposition is determined. Each measured survey defines a survey positionin the wellbore and an uncertainty of the survey position.

Another aspect of the invention relates to a method for estimating aposition in a wellbore. The method involves taking a plurality ofsurveys of the wellbore and combining overlapping portions of thesurveys whereby the wellbore position is determined. Each measuredsurvey defines a survey position in the wellbore and an uncertainty ofthe survey position.

Another aspect of the invention relates to a method for estimating aposition in a wellbore. The method involves acquiring a plurality ofsurveys of the wellbore and combining overlapping portions of thesurveys whereby the wellbore position is determined. Each measuredsurvey defines a survey position in the wellbore and an uncertainty ofthe survey position. The surveys are combined using the followingequation: MPP=((H_(n) ^(T)Cov_(n) ⁻¹H_(n))⁻¹H_(n) ^(T)Cov_(n) ⁻¹)*V.

Other aspects and advantages of the invention will be apparent from thefollowing description and the appended claims.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a schematic view of a drilling rig having a drilling apparatusextending into a wellbore penetrating a subterranean formation to surveythe wellbore;

FIG. 2 is a schematic view of the wellbore of FIG. 4 having a wirelinetool positioned therein to survey the wellbore;

FIG. 3 is a graphic depiction of survey points along a path and theirassociated ellipsoids of uncertainty;

FIG. 4 is graphic depiction of two surveys and related uncertainties ata position along a path combined to estimate a resultant position andresultant uncertainty;

FIG. 5 is a cross-sectional view of the graphic depiction of FIG. 4taken along line 5—5;

FIG. 6 is a schematic view of the wellbore of FIG. 1 depicts a resultantposition determined from overlapping estimated survey positions andrelated ellipsoids of uncertainty at position r_(VII) in the wellbore;and

FIG. 7 is a schematic view of the wellbore of FIG. 6 extended a distancefurther into the subterranean formation and depicting a resultantposition determined from overlapping portions of estimated surveypositions and related ellipsoids of uncertainty.

DETAILED DESCRIPTION

Illustrative embodiments of the invention are described below. In theinterest of clarity, not all features of an actual implementation aredescribed in this specification. It will of course be appreciated thatin the development of any such actual embodiment, numerousimplementation-specific decisions must be made to achieve thedevelopers' specific goals, such as compliance with system-related andbusiness-related constraints, which will vary from one implementation toanother. Moreover, it will be appreciated that such a developmenteffort, even if complex and time-consuming, would be a routineundertaking for those of ordinary skill in the art having the benefit ofthis disclosure.

Referring now to the drawings in general and FIG. 1 in particular, anenvironment in which the present invention may be utilized is depicted.FIG. 1 shows drilling rig 10 having a drilling tool 12 extendingdownhole into a wellbore 14 penetrating a subterranean formation 15. Thedrilling tool 12 extends from the surface 16 at known position r₀ to thebottom 18 of the wellbore 14 at estimated survey position r_(VII).Incremental survey positions r_(I) through r_(VI) extend between r₀ andr_(VII). Incremental survey positions r_(I) through r_(VII) areestimated and/or measured using one or more of the known surveytechniques.

The drilling tool 12 depicted in FIG. 1 is capable of collecting surveydata and other information while the drilling tool drills the wellboreusing known survey techniques. The drilling tool 12 may be used tosurvey and/or collect data before, during or after a drilling operation.The measurements taken using the drilling tool may be done continuouslyand/or at discrete positions in the wellbore. The drilling tool 12 isalso capable of surveying and/or collecting data as the tool is extendeddownhole and/or retrieved uphole in a continuous and/or discrete manner.The drilling tool 12 is capable of taking a survey along one or more ofthe survey points r₀ through r_(VII).

Referring now to FIG. 2, the drilling rig 10 of FIG. 1 is shown with awireline tool 20 extending into the wellbore 14. The wireline tool 20 islowered into the wellbore 14 to survey and/or collect data. The wirelinetool 20 is capable of surveying and/or collecting data as the tool isextended downhole and/or retrieved uphole in a continuous and/ordiscrete manner. As with the drilling tool, the wireline tool is alsocapable of taking a survey along one or more of the survey points r₀through r_(VII) as the tool is advanced uphole and/or downhole.

As shown in FIGS. 1 and 2, various tools may be used to take one or moresurveys (individually and/or collectively) in a continuous and/ordiscrete manner as will be appreciated by one skilled in the art. Forsimplicity, a curved wellbore is shown; however, the wellbore may be ofany size or shape, vertical, horizontal and/or curved. Additionally, thewellbore may be a land unit as shown, or an offshore well.

The estimated survey positions and related positional uncertaintyassociated with surveys is mathematically depicted in as shown in FIG.3. FIG. 3 represents a plurality of surveys taken along a wellborebeginning at a known reference position r₀ and terminating at anestimated survey position r_(VII), with estimated survey positions r_(I)through r_(VI) therebetween. The position of survey positions r_(I)through r_(VII) is estimated using known survey techniques. As depictedin FIG. 3, estimated survey positions r_(I) through r_(VII) areprogressively further away from known reference position r₀. Theestimated survey positions r_(I) through r_(VII) may be connected toform an estimated trajectory 22 using known survey techniques.

Because r₀ is known, it is presumed to have little or no uncertainty. Asdepicted in FIG. 3, the estimated position of each survey point r_(I)through r_(VII) has an “ellipsoid of uncertainty” E₁ through E₇surrounding a corresponding survey point, respectively. Each ellipsoidsE represent the uncertainty associated with its respective position.

Where overlapping surveys are taken along a wellbore, they may becombined, as visually depicted in FIG. 4. A first survey is taken from aknown position r₀ to an estimated position r_(VII). With respect to FIG.4, a first trajectory 22 a beginning at an known position 25 a andextending to an estimated survey position 30 a having an ellipsoid ofuncertainty 24 a is shown. A second trajectory 22 b beginning at knownposition 25 a and extending to an estimated survey position 30 b havingan ellipsoid of uncertainty 24 b is also shown. First survey position 30a and its first ellipsoid of uncertainty 24 a is combined with secondsurvey position 30 b and its second ellipsoid of uncertainty 24 b toform a resultant position 28 a. Similarly, first ellipsoid ofuncertainty 24 a is combined with second ellipsoid of uncertainty 24 bto form a resultant ellipsoid of uncertainty 26 a. For further clarity,a cross-sectional view of FIG. 4 taken along line 5—5 is depicted inFIG. 5.

The combination of the survey positions r may also be represented bymathematical calculations. Overlapping estimated survey positions may becharacterized in the form of a position vector V. Position vector Vcontains position vectors r for each of n overlapping surveys performedat a position in a wellbore. Each position vector r has an x, y and zcoordinate representing a survey position estimated by known surveytechniques. The position vector V combines the position vectors r toform the stacked 3n×1 vector V below: $V = {\begin{matrix}r_{1x} \\r_{1y} \\r_{1z} \\r_{2x} \\r_{2y} \\r_{2z} \\\vdots \\r_{nx} \\r_{ny} \\r_{nz}\end{matrix}}$

The ellipsoid of uncertainty for each estimated survey position vector rhaving an (x, y and z) coordinate, is mathematically represented by thecovariance matrix (Cov_(r)) set forth below, and the combination of theCov_(r) matrices for n overlapping surveys is mathematically representedby the 3n×3n covariance matrix (Cov_(n)) set forth below:$\begin{matrix}{{Cov}_{r} = \begin{bmatrix}{\langle{\delta \quad r_{x}\delta \quad r_{x}}\rangle} & {\langle{\delta \quad r_{x}\delta \quad r_{y}}\rangle} & {\langle{\delta \quad r_{x}\delta \quad r_{z}}\rangle} \\{\langle{\delta \quad r_{y}\delta \quad r_{x}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{y}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{z}}\rangle} \\{\langle{\delta \quad r_{z}\delta \quad r_{x}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{z}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{z}}\rangle}\end{bmatrix}} \\{{Cov}_{n} = \begin{bmatrix}{\langle{\delta \quad {r1}_{x}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {r1}_{x}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {r1}_{y}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {r1}_{y}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {r1}_{x}}\rangle} & \cdots & {\langle{\delta \quad {r1}_{y}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {r1}_{y}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {r1}_{y}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {r1}_{z}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {r1}_{z}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {rn}_{z}}\rangle} \\\vdots & \vdots & \vdots & ⋰ & \vdots & \vdots & \vdots \\{\langle{\delta \quad {rn}_{x}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {rn}_{x}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {rn}_{y}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {rn}_{z}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {rn}_{z}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {rn}_{z}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {rn}_{z}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {rn}_{z}\delta \quad {rn}_{z}}\rangle}\end{bmatrix}}\end{matrix}$

This 3n×3n matrix (Cov_(n)) defines the auto and cross covariancebetween associated estimated survey positions (r). The covariancerepresents the statistical relationship between the estimated surveypositions. The resultant position of the combined surveys, or “MostProbable Position (MPP)”, may then be calculated using the followingequation:

MPP=((H _(n) ^(T)Cov_(n) ⁻¹ H _(n))⁻¹ H _(n) ^(T)Cov_(n) ⁻¹)*V

Where H is the 3×3 identity matrix, H_(n) consists of n3×3 identitymatrices stacked up where n is number of overlapping surveys and HUT isthe transpose of H_(n) as set forth below: $H = {{{\begin{matrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{matrix}}\quad H_{n}} = {\begin{matrix}1_{1} & 0 & 0 \\0 & 1_{1} & 0 \\0 & 0 & 1_{1} \\1_{2} & 0 & 0 \\0 & 1_{2} & 0 \\0 & 0 & 1_{2} \\\vdots & \vdots & \vdots \\1_{n} & 0 & 0 \\0 & 1_{n} & 0 \\0 & 0 & 1_{n}\end{matrix}}}$ $H_{n}^{T} = {\begin{matrix}1_{1} & 0 & 0 & 1_{2} & 0 & 0 & \cdots & 1_{n} & 0 & 0 \\0 & 1_{1} & 0 & 0 & 1_{2} & 0 & \cdots & 0 & 1_{n} & 0 \\0 & 0 & 1_{1} & 0 & 0 & 1_{2} & \cdots & 0 & 0 & 1_{n}\end{matrix}}$

The corresponding resultant positional uncertainty for the resultantposition (MPP) is defined by a covariance matrix represented by thefollowing equation:

Cov_(MPP)=(H _(n) ^(T)Cov_(n) ⁻¹ H _(n))⁻¹

The resultant position (MPP) and corresponding resultant positionaluncertainty(Cov_(MPP)) represent the position and uncertainty for noverlapping surveys having been combined using this technique.

Applying the mathematical model to wellbore operations, the surveys andellipsoids of uncertainty for multiple overlapping surveys of a wellboreare depicted in FIG. 6. Each survey performed along the wellboregenerates data indicating the survey position of the wellbore with itsrelated ellipsoid of uncertainty at points r₀ through r_(VII). FIG. 6depicts a first trajectory 22 e taken along wellbore 14 using thedrilling tool of FIG. 1, and a second trajectory 22 f taken alongwellbore 14 using the wireline tool of FIG. 2. At wellbore positionr_(VII), the first trajectory terminates at a first survey position 30 ehaving an ellipsoid of uncertainty 24 e, and second trajectoryterminates at a second survey position 30 f having a second ellipsoid ofuncertainty 24 f. The first and second survey positions 30 e and 30 fand their corresponding first and second ellipsoids of uncertainty 24 eand 24 f are combined to generate a resultant position (MPP) 28 c andcorresponding resultant ellipsoid of uncertainty 26 c.

While FIG. 6 depicts two overlapping surveys combined to generate theresultant position and related ellipsoid of uncertainty, it will beappreciated that multiple overlapping surveys may be combined togenerate the resultant position (MPP) and related resultant uncertainty.Applying the mathematical principles to the wellbore operation set forthin FIG. 6, the resultant position of the wellbore at point r_(VII) maybe estimated. During the wellbore operation of a section of the wellbore14, surveys are recorded along a wellpath using known survey techniquesresulting in an estimated survey position along the wellpath. Thesesurveys positions are generally referenced to a measured or assigneddepth, or distance along the wellpath from a known surface location.

During wellbore operations, various survey measurements produce one ormore overlapping estimated survey positions along the wellpath. Thistechnique can then be applied to combine any number of overlappingsurvey measurements at the same wellbore position for any interval overthe wellpath for which such multiple survey measurements exist.

For example, the first survey 22 e may produce a survey position 30 erepresented by r₁ (x,y,z)=(10,10,100), and the second survey 22 f mayproduce survey position 30 f represented by r₂ (x,y,z)=(−10,−10,120).These measurements may be translated into the following position vector:

V=[10;10;100;−10;−10;120]

In this example, each of the overlapping estimated survey positions hasa given uncertainty represented by Cov₁ and Cov₂ as depicted in thecovariant matrix below:

Cov₁ and Cov₂=[100,0,0;0,169,0;0,0,25]

The Cov₁ and Cov₂ matrix generates the following covariance matrix:${Cov}_{n} = \begin{matrix}100 & 0 & 0 & 0 & 0 & 0 \\0 & 169 & 0 & 0 & 0 & 0 \\0 & 0 & 25 & 0 & 0 & 0 \\0 & 0 & 0 & 100 & 0 & 0 \\0 & 0 & 0 & 0 & 169 & 0 \\0 & 0 & 0 & 0 & 0 & 25\end{matrix}$

The first and second overlapping surveys may be combined to generate theMPP as follows:

MPP=((H _(n) ^(T)Cov_(n) ⁻¹ H _(n))⁻¹ H _(n) ^(T)Cov_(n) ⁻¹)*V

MPP=0,0,110

where:

H _(n)=[1 0 0;0 1 0;0 0 1;1 0 0;0 1 0;0 0 1]

and n=2

In this example, the resultant position vector is equidistant betweenthe two survey points as expected for this example. The covariancematrix may then be solved as follows: $\begin{matrix}{{Cov}_{MPP} = \left( {H_{n}^{T}{Cov}_{n}^{- 1}H_{n}} \right)^{- 1}} \\{= \begin{matrix}50 & 0 & 0 \\0 & 84.5 & 0 \\0 & 0 & 12.5\end{matrix}}\end{matrix}$

The result of this process is then a resultant position 28 c (MPP) basedon combining overlapping surveys at the same position r_(VII) in thewellbore.

For simplicity, this example incorporated positions with identicalcovariance matrices; however, it will be appreciated that differentsurveys may have different covariance matrices.

Referring now to FIG. 7, the wellbore 14 of FIG. 1 is drilled furtherinto formation 15. The wellbore 14 extends beyond original bottom 18 atposition r_(VII) to new bottom 32 at position r_(X). A new survey istypically taken during the subsequent drilling operation for theextended wellbore 14,′ or by a wireline tool. The portion 22 g of thenew survey of wellbore 14′ along points r₀ to r_(VII) may be combinedwith existing surveys of the original wellbore 14 (FIGS. 1, 2 and 6)from overlapping positions r₀ to r_(VII) as heretofore described. Theestimated survey positions 30 e and 30 g at position r_(VII) in thewellbore and related ellipsoids of uncertainty 24 e and 24 g,respectively, may be combined as heretofore described to generateresultant position (MPP) 28 d and related ellipsoid of uncertainty 26 d.The portion 22 g′ of the new survey of wellbore 14′ along pointr_(VIII to r) _(X) has an estimated survey position 30 g′ and relatedellipsoid of uncertainty 24 g′.

The resultant position 28 d may then be used to calculate a resultantposition 28 d′ at wellbore position r_(X) using known survey techniques.This can be expressed as the equation:

28 d′=28 d +(28 d′−28 d)

The ellipsoid of uncertainty 26 d′ for resultant position 28 d′ may thenbe estimated using known techniques by applying the following equation:⟨δ  28d^(′)  δ  28d^(′  tr)⟩ = ⟨δ  28d  δ  28d^(tr)⟩ + ⟨(δ  28d^(′)−  δ  28d)(δ  28d^(′)−  δ  28d)^(tr)⟩⟨δ  28d(δ  28d^(′)−  δ  28d)^(tr)⟩ + ⟨(δ  28d^(′)−  δ  28d)  δ  28d^(tr)⟩

While the invention has been described with respect to a limited numberof embodiments, those skilled in the art, having benefit of thisdisclosure, will appreciate that other embodiments can be devised whichdo not depart from the scope of the invention as disclosed herein.Accordingly, the scope of the invention should be limited only by theattached claims.

What is claimed is:
 1. A method for estimating a position of a wellbore,comprising: acquiring a plurality of surveys of the wellbore, eachsurvey defining a survey position of the wellbore and an uncertainty ofthe survey position; and combining the uncertainties of the surveypositions whereby the wellbore position is determined.
 2. The method ofclaim 1 wherein in the step of acquiring, at least one survey is takenwhile drilling the wellbore.
 3. The method of claim 2 wherein in thestep of acquiring, at least one survey is taken using a wireline tool.4. The method of claim 1 further comprising the step of extending thewellbore a distance further thereby defining an extended wellbore, andwherein in the step of acquiring, at least a portion of at least onesurvey is taken of the extended wellbore.
 5. The method of claim 4further comprising estimating a position in the extended wellbore usingthe wellbore position.
 6. The method of claim 1 wherein in the step ofacquiring, at least one survey is taken using a wireline tool.
 7. Themethod of claim 1 wherein in the step of combining, the wellboreposition is estimated using the following equation: MPP=((H _(n)^(T)Cov_(n) ⁻¹ H _(n))⁻¹ H _(n) ^(T)Cov_(n))*V where$H_{n} = {{{\begin{matrix}1_{1} & 0 & 0 \\0 & 1_{1} & 0 \\0 & 0 & 1_{1} \\1_{2} & 0 & 0 \\0 & 1_{2} & 0 \\0 & 0 & 1_{2} \\\vdots & \vdots & \vdots \\1_{n} & 0 & 0 \\0 & 1_{n} & 0 \\0 & 0 & 1_{n}\end{matrix}}\quad H_{n}^{T}} = {\begin{matrix}1_{1} & 0 & 0 & 1_{2} & 0 & 0 & \cdots & 1_{n} & 0 & 0 \\0 & 1_{1} & 0 & 0 & 1_{2} & 0 & \cdots & 0 & 1_{n} & 0 \\0 & 0 & 1_{1} & 0 & 0 & 1_{2} & \cdots & 0 & 0 & 1_{n}\end{matrix}}}$ ${Cov}_{n} = \begin{bmatrix}{\langle{\delta \quad {r1}_{x}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad r_{z}}\rangle} & \cdots & {\langle{\delta \quad {r1}_{x}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {r1}_{y}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {r1}_{y}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {r1}_{x}}\rangle} & \cdots & {\langle{\delta \quad {r1}_{y}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {r1}_{y}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {r1}_{y}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {r1}_{z}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {r1}_{z}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {rn}_{z}}\rangle} \\\vdots & \vdots & \vdots & ⋰ & \vdots & \vdots & \vdots \\{\langle{\delta \quad {rn}_{x}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {rn}_{x}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {rn}_{y}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {rn}_{z}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {rn}_{z}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {rn}_{z}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {rn}_{z}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {rn}_{z}\delta \quad {rn}_{z}}\rangle}\end{bmatrix}$ $V = {{{\begin{matrix}r_{1x} \\r_{1y} \\r_{1z} \\r_{2x} \\r_{2y} \\r_{2z} \\\vdots \\r_{nx} \\r_{ny} \\r_{nz}\end{matrix}}\quad {Cov}_{r}} = \begin{bmatrix}{\langle{\delta \quad r_{x}\delta \quad r_{x}}\rangle} & {\langle{\delta \quad r_{x}\delta \quad r_{y}}\rangle} & {\langle{\delta \quad r_{x}\delta \quad r_{z}}\rangle} \\{\langle{\delta \quad r_{y}\delta \quad r_{x}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{y}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{z}}\rangle} \\{\langle{\delta \quad r_{z}\delta \quad r_{x}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{z}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{z}}\rangle}\end{bmatrix}}$

r=the position of each survey point (1−n) having (x,y,z) coordinatesn=the number of surveys taken.
 8. The method of claim 7 wherein theresultant uncertainty is calculated from the equation: Cov_(MPP)=(H _(n)^(T)Cov_(n) ⁻¹ H _(n))⁻¹.
 9. A method for estimating a position of awellbore, comprising: drilling a wellbore into a subterranean formation;acquiring a plurality of surveys of the wellbore, each survey defining asurvey position of the wellbore and an uncertainty of the surveyposition; and combining the uncertainties of the survey position wherebythe wellbore position is determined.
 10. The method of claim 9 whereinin the step of acquiring, at least one survey is taken while drillingthe wellbore.
 11. The method of claim 10 wherein in the step ofacquiring, at least one survey is taken using a wireline tool.
 12. Themethod of claim 9 further comprising the step of extending the wellborea distance further thereby defining an extended wellbore, and wherein inthe step of acquiring, at least a portion of at least one survey istaken of the extended wellbore.
 13. The method of claim 12 furthercomprising estimating a position in the extended wellbore using thewellbore position.
 14. The method of claim 9 wherein in the step ofacquiring, at least one survey is taken using a wireline tool.
 15. Themethod of claim 9 wherein in the step of combining, the wellboreposition is estimated using the following equation: MPP=((H _(n)^(T)Cov_(n) ⁻¹ H _(n))⁻¹ H _(n) ^(T)Cov_(n) ⁻¹)*V where$H_{n} = {{{\begin{matrix}1_{1} & 0 & 0 \\0 & 1_{1} & 0 \\0 & 0 & 1_{1} \\1_{2} & 0 & 0 \\0 & 1_{2} & 0 \\0 & 0 & 1_{2} \\\vdots & \vdots & \vdots \\1_{n} & 0 & 0 \\0 & 1_{n} & 0 \\0 & 0 & 1_{n}\end{matrix}}\quad H_{n}^{T}} = {\begin{matrix}1_{1} & 0 & 0 & 1_{2} & 0 & 0 & \cdots & 1_{n} & 0 & 0 \\0 & 1_{1} & 0 & 0 & 1_{2} & 0 & \cdots & 0 & 1_{n} & 0 \\0 & 0 & 1_{1} & 0 & 0 & 1_{2} & \cdots & 0 & 0 & 1_{n}\end{matrix}}}$ ${Cov}_{n} = \begin{bmatrix}{\langle{\delta \quad {r1}_{x}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad r_{z}}\rangle} & \cdots & {\langle{\delta \quad {r1}_{x}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {r1}_{y}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {r1}_{y}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {r1}_{x}}\rangle} & \cdots & {\langle{\delta \quad {r1}_{y}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {r1}_{y}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {r1}_{y}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {r1}_{z}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {r1}_{z}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {rn}_{z}}\rangle} \\\vdots & \vdots & \vdots & ⋰ & \vdots & \vdots & \vdots \\{\langle{\delta \quad {rn}_{x}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {rn}_{x}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {rn}_{y}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {rn}_{z}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {rn}_{z}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {rn}_{z}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {rn}_{z}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {rn}_{z}\delta \quad {rn}_{z}}\rangle}\end{bmatrix}$ $V = {{{\begin{matrix}r_{1x} \\r_{1y} \\r_{1z} \\r_{2x} \\r_{2y} \\r_{2z} \\\vdots \\r_{nx} \\r_{ny} \\r_{nz}\end{matrix}}\quad {Cov}_{r}} = \begin{bmatrix}{\langle{\delta \quad r_{x}\delta \quad r_{x}}\rangle} & {\langle{\delta \quad r_{x}\delta \quad r_{y}}\rangle} & {\langle{\delta \quad r_{x}\delta \quad r_{z}}\rangle} \\{\langle{\delta \quad r_{y}\delta \quad r_{x}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{y}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{z}}\rangle} \\{\langle{\delta \quad r_{z}\delta \quad r_{x}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{z}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{z}}\rangle}\end{bmatrix}}$

r=the position of each survey point (1−n) having (x,y,z) coordinatesn=the number of surveys taken.
 16. The method of claim 15 wherein theresultant uncertainty is calculated from the equation: Cov_(MPP)=(H _(n)^(T)Cov_(n) ⁻¹ H _(n))⁻¹.
 17. A method for estimating a position of awellbore, comprising: taking a plurality of surveys of the wellbore,each survey defining a survey position of the wellbore and anuncertainty of the survey position; and combining the uncertainties ofthe survey positions whereby the wellbore position is determined. 18.The method of claim 17 wherein in the step of acquiring, at least onesurvey is taken while drilling the wellbore.
 19. The method of claim 18wherein in the step of acquiring, at least one survey is taken using awireline tool.
 20. The method of claim 17 further comprising the step ofextending the wellbore a distance further thereby defining an extendedwellbore, and wherein in the step of acquiring, at least a portion of atleast one survey is taken of the extended wellbore.
 21. The method ofclaim 20 further comprising estimating a position in the extendedwellbore using the wellbore position.
 22. The method of claim 17 whereinin the step of acquiring, at least one survey is taken using a wirelinetool.
 23. The method of claim 17 wherein in the step of combining, thewellbore position is estimated using the following equation: MPP=((H_(n) ^(T)Cov_(n) ⁻¹ H _(n))⁻¹ H _(n) ^(T)Cov_(n) ⁻¹)*V where$H_{n} = {{{\begin{matrix}1_{1} & 0 & 0 \\0 & 1_{1} & 0 \\0 & 0 & 1_{1} \\1_{2} & 0 & 0 \\0 & 1_{2} & 0 \\0 & 0 & 1_{2} \\\vdots & \vdots & \vdots \\1_{n} & 0 & 0 \\0 & 1_{n} & 0 \\0 & 0 & 1_{n}\end{matrix}}\quad H_{n}^{T}} = {\begin{matrix}1_{1} & 0 & 0 & 1_{2} & 0 & 0 & \cdots & 1_{n} & 0 & 0 \\0 & 1_{1} & 0 & 0 & 1_{2} & 0 & \cdots & 0 & 1_{n} & 0 \\0 & 0 & 1_{1} & 0 & 0 & 1_{2} & \cdots & 0 & 0 & 1_{n}\end{matrix}}}$ ${Cov}_{n} = \begin{bmatrix}{\langle{\delta \quad {r1}_{x}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad r_{z}}\rangle} & \cdots & {\langle{\delta \quad {r1}_{x}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {r1}_{y}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {r1}_{y}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {r1}_{x}}\rangle} & \cdots & {\langle{\delta \quad {r1}_{y}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {r1}_{y}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {r1}_{y}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {r1}_{z}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {r1}_{z}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {rn}_{z}}\rangle} \\\vdots & \vdots & \vdots & ⋰ & \vdots & \vdots & \vdots \\{\langle{\delta \quad {rn}_{x}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {rn}_{x}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {rn}_{y}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {rn}_{z}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {rn}_{z}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {rn}_{z}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {rn}_{z}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {rn}_{z}\delta \quad {rn}_{z}}\rangle}\end{bmatrix}$ $V = {{{\begin{matrix}r_{1x} \\r_{1y} \\r_{1z} \\r_{2x} \\r_{2y} \\r_{2z} \\\vdots \\r_{nx} \\r_{ny} \\r_{nz}\end{matrix}}\quad {Cov}_{r}} = \begin{bmatrix}{\langle{\delta \quad r_{x}\delta \quad r_{x}}\rangle} & {\langle{\delta \quad r_{x}\delta \quad r_{y}}\rangle} & {\langle{\delta \quad r_{x}\delta \quad r_{z}}\rangle} \\{\langle{\delta \quad r_{y}\delta \quad r_{x}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{y}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{z}}\rangle} \\{\langle{\delta \quad r_{z}\delta \quad r_{x}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{z}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{z}}\rangle}\end{bmatrix}}$

r=the position of each survey point (1−n) having (x,y,z) coordinatesn=the number of surveys taken.
 24. The method of claim 23 wherein theresultant uncertainty is calculated from the equation: Cov_(MPP)=(H _(n)^(T)Cov_(n) ⁻¹ H _(n))⁻¹.
 25. A method for estimating a position in awellbore, comprising: acquiring a plurality of surveys of the wellbore,each survey defining a survey position in the wellbore and anuncertainty of the survey position; and combining the uncertainties ofthe survey positions whereby the wellbore position is determined usingthe following equation: MPP=((H _(n) ^(T)Cov_(n) ⁻¹ H _(n))⁻¹ H _(n)^(T)Cov_(n) ⁻¹)*V where $H_{n} = {{{\begin{matrix}1_{1} & 0 & 0 \\0 & 1_{1} & 0 \\0 & 0 & 1_{1} \\1_{2} & 0 & 0 \\0 & 1_{2} & 0 \\0 & 0 & 1_{2} \\\vdots & \vdots & \vdots \\1_{n} & 0 & 0 \\0 & 1_{n} & 0 \\0 & 0 & 1_{n}\end{matrix}}\quad H_{n}^{T}} = {\begin{matrix}1_{1} & 0 & 0 & 1_{2} & 0 & 0 & \cdots & 1_{n} & 0 & 0 \\0 & 1_{1} & 0 & 0 & 1_{2} & 0 & \cdots & 0 & 1_{n} & 0 \\0 & 0 & 1_{1} & 0 & 0 & 1_{2} & \cdots & 0 & 0 & 1_{n}\end{matrix}}}$ ${Cov}_{n} = \begin{bmatrix}{\langle{\delta \quad {r1}_{x}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad r_{z}}\rangle} & \cdots & {\langle{\delta \quad {r1}_{x}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {r1}_{y}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {r1}_{y}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {r1}_{x}\delta \quad {r1}_{x}}\rangle} & \cdots & {\langle{\delta \quad {r1}_{y}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {r1}_{y}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {r1}_{y}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {r1}_{z}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {r1}_{z}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {r1}_{z}\delta \quad {rn}_{z}}\rangle} \\\vdots & \vdots & \vdots & ⋰ & \vdots & \vdots & \vdots \\{\langle{\delta \quad {rn}_{x}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {rn}_{x}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {rn}_{x}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {rn}_{y}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {rn}_{z}}\rangle} \\{\langle{\delta \quad {rn}_{z}\delta \quad {r1}_{x}}\rangle} & {\langle{\delta \quad {rn}_{z}\delta \quad {r1}_{y}}\rangle} & {\langle{\delta \quad {rn}_{y}\delta \quad {r1}_{z}}\rangle} & \cdots & {\langle{\delta \quad {rn}_{z}\delta \quad {rn}_{x}}\rangle} & {\langle{\delta \quad {rn}_{z}\delta \quad {rn}_{y}}\rangle} & {\langle{\delta \quad {rn}_{z}\delta \quad {rn}_{z}}\rangle}\end{bmatrix}$ $V = {{{\begin{matrix}r_{1x} \\r_{1y} \\r_{1z} \\r_{2x} \\r_{2y} \\r_{2z} \\\vdots \\r_{nx} \\r_{ny} \\r_{nz}\end{matrix}}\quad {Cov}_{r}} = \begin{bmatrix}{\langle{\delta \quad r_{x}\delta \quad r_{x}}\rangle} & {\langle{\delta \quad r_{x}\delta \quad r_{y}}\rangle} & {\langle{\delta \quad r_{x}\delta \quad r_{z}}\rangle} \\{\langle{\delta \quad r_{y}\delta \quad r_{x}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{y}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{z}}\rangle} \\{\langle{\delta \quad r_{z}\delta \quad r_{x}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{z}}\rangle} & {\langle{\delta \quad r_{y}\delta \quad r_{z}}\rangle}\end{bmatrix}}$

r=the position of each survey point (1−n) having (x,y,z) coordinatesn=the number of surveys taken.
 26. The method of claim 25 wherein theresultant uncertainty is calculated from the equation: Cov_(MPP)=(H _(n)^(T) Cov_(n) ⁻¹ H _(n))⁻¹.